Ординатура / Офтальмология / Английские материалы / Principles Of Medical Statistics_Feinstein_2002

in Section 16.1.2.1, however, rest on the assumptions that the data are Gaussian, with equal sizes and variances in both groups. Nevertheless, the results will not be too disparate from these calculations if those assumptions do not hold. In general, the contrast will almost always be stochastically significant at 2P < .05 if the two boxes have no overlap at all or if they have no more than a Half-H-spread overlap for group sizes ≥ 20.

16.1.2.3 Published Examples — Because the box plot is a relatively new form of display, its full potential as a visual aid has not yet been fully exploited for stochastic decisions. For scientists who want descriptive communication and who would prefer to decide about “significance” by directly examining quantitative distinctions in “bulks” of data, a contrast of box plots offers an excellent way to compare the spreads of data and the two bulks demarcated by the interquartile zones. The comparative display of two box plots can thus be used not only to summarize data for the two groups, but also to allow “stochastic significance” to be appraised from inspections of bulk, not merely from the usual P values and confidence intervals. Because box-plot displays have been so uncommon, however, few illustrations are available from published literature.

Figure 16.10 shows such a display, which was presented in a letter to the editor. 4 The spread is completely shown for the data, which are not Gaussian, because the boxes and ranges are not symmetrical around the median (marked with an asterisk rather than a horizontal line). Stochastic significance can promptly be inferred for this contrast because the two boxes are almost completely separated. The only disadvantage of this display (and of the associated text of the letter) is that the group sizes are not mentioned.

Figure 16.11 shows another contrast of box plots, obtained during a previously mentioned (Chapter 10) study, by Burnand et al.,5 of investigators’ deci-

sions regarding quantitative significance for a ratio of two means. The

group sizes are indicated; and the medians are drawn with lines (rather than

asterisks). The “whiskers” for each group cover an ipr95, rather than the

“fence” and “outlier” tactic proposed by Tukey. With almost no overlap in the boxes, the obvious differences in the groups will also be stochastically significant.

In a study of cardiac size on chest films, shown in Figure 16.12, Nickol et al.6 presented box-plot contrasts for three variables in three ethnic groups. The investigators stated that “every comparison (in these data) differed significantly (p < 0.001).” Using the Half- H-spread exclusion principle, the box plots for the investigated groups show

obviously significant differences between Africans and Asians in cardiac diameter, between Africans and Caucasians in cardiothoracic ratio, and between Africans and Caucasians in age.

16.1.2.4 Indications of Group Size — One disadvantage of box plots is that group sizes are not routinely indicated. The sizes can easily be inserted, however, at the top of each “whisker,” as shown in Figure 16.11. Some writers have recommended7 that the width of the boxes be adjusted to

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reflect the size of each group. Since unequal widths might impair the discernment of lengths in the boxes, the group sizes are probably best specified with a simple indication of N for each group.

Despite its use as a descriptive summary of data, the box-plot contrast discussed throughout Section 16.1.2 was employed for stochastic decisions. An elegant way of contrasting the quantitative distinctions of two groups, regardless of their sizes, has been devised by Wilk and Granadesikan.9 They originally proposed “quantile (Q-Q) plots, percent (P-P) plots, and hybrids of these” to compare an observed univariate group with a proposed theoretical distribution for the group, but the tactic can be used to contrast distributions for two different groups. The quantile technique, which relies on quintiles, quartiles, medians, etc., is probably easier to apply than the percent P-P plot, which relies on cumulative percentiles.

The quantile-quantile (Q-Q) graph creates a “pairing,” otherwise unattainable with nonpaired data, in which the respective X and Y values at each point are the corresponding quantiles of the compared Groups A and B. The two groups

can then be compared descriptively according to the line connecting the Q-Q points. If points lie along the “identity” line X = Y, the two groups have similar results. If the line goes through the origin with slope 1.2, the values in one group are in general 20% higher than in the other.

The method has seldom been used in medical literature, so the illustration here has another source. In Figure 16.14, the ozone concentrations for Stamford vs. Yonkers are plotted with a large array of quantiles, such as 2nd percentile, 5th percentile, 10th percentile, quartile, median, for each group. A much simpler line, however, could be drawn for just three points: the median and the lower and upper quartiles in each group. Four additional points — the lower and upper quadragintile values and the 10and 90-percentile values might be added for more complete detail.

C.P. Jones10 has pointed out that the relationship of lines and crossings in the Q-Q plot can be used to demonstrate descriptive differences in location, spread, and shape for the two compared groups of data. Jones has also proposed a simple “projection plot” in which the difference in each pair of quantile values is plotted against their average value. With this arrangement, the identity line passes horizontally through the increment of 0.

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FIGURE 16.12

Box-and-whisker plots of the distribution of the data for radiographic cardiac diameter, cardiothoracic ratio, and age, by ethnic groups; AFR = African, ASI = Asian, CAU = caucasian. The box encloses the interquartile range, and is transected by a heavy bar at the median. The whiskers encompass the 25% of data beyond the interquartile range in each direction (after Tukey, 1977). Skew was statistically significant in 5 groups of data marked (S), p < 0.05, but was of very moderate degree in all but the data for Age of caucasian subjects. Age in African subjects showed kurtosis marked (K), p < 0.05, of mild degree. [Figure and legend taken from Chapter Reference 6.]

FIGURE 16.13

Distributions of CD4+ lymphocyte counts in injecting drug users and homosexual men, with each cohort divided into individuals who did or did not develop acquired immunodeficiency syndrome (AIDS) or thrush, as described in the text. Boxes indicate median and upper and lower hinges; “waists” around the medians indicate 95% confidence intervals of the medians. The lowermost and uppermost horizontal lines indicate fences, either 1.5 interquartile ranges away from the first and third quartiles or at the limit of the observations, whichever was smaller. Outliers (values beyond the fences) are marked by asterisks. [Figure and legend taken from Chapter Reference 8.]

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16.2 Principles of Display for Dimensional Variables

The main challenges in displaying individual points of dimensional data are choices of symbols, methods of portraying multiple points at the same location, and selection of summary indexes.

16.2.1.1Symbols for Points — In most instances, dimensional values are shown in a vertical array within the arbitrary horizontal width assigned for columns of each compared group. Because each group usually occupies its own column, different symbols are seldom needed for the points of different groups.

A simple, circular, filled-in dot, enlarged just enough to be clearly visible, is preferable to open circles, squares, triangles, crosses, diamonds, and other shapes. The other shapes become useful, however, to distinguish one group from another when two or more groups are contrasted within the same vertical column.

16.2.1.2Multiple Points at Same Location — Multiple points at the same vertical location can be spread out in a horizontal cluster as ……, and the arbitrary width of the columns can be adjusted

as needed. If the spread becomes too large, the points can be split and placed close together in a double or triple tier. A remarkable “cannonball” spread of points is shown for multiple groups12 in Figure 16.15.

Instead of appearing individually in a cluster (or “cannonball”) arrangement, the multiple points are sometimes combined into a single expanded symbol, such as a large circle, whose area must appropriately represent the individual frequencies. The area is easy to identify if the symbol is a square, but is difficult to appraise if it must be calculated for a circle or other non-square symbol. Enlarged symbols can be particularly useful in a two-dimensional graph, whose spacing cannot be arbitrarily altered to fit the

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FIGURE 16.15

Serum-ascites albumin gradient in ascites, classified by presence or absence of portal hypertension. Statistical comparisons are to the sterile cirrhotic group by unpaired t-test. CARD = cardiac ascites; Misc Non-PHT = miscellaneous- nonportal-hypertension-related; Misc PHT = miscellaneous portal-hypertension-related; NS = not significant; PCA = peritoneal carcinomatosis. Mean ± SD bars are included as well as a horizontal line at 11 g/L, the threshold for portal hypertension. All groups differed significantly (P < 0.05) from the sterile cirrhotic samples by analysis of variance with the Dunnette test. [Figure and legend taken from Chapter Reference 12.]

individual cluster. In comparisons of categorical groups, however, the enlarged symbols usually keep the viewer from discerning individual frequency counts.

16.2.1.3 Choice of Summary Symbols — The decisions about summary symbols for indexes of central tendency, spread, and size depend on whether the purpose of the display is descriptive communication or stochastic inference. Many displays seem to be aimed at stochastic decisions, showing bars and flanges for means and standard errors but not points of data. Because the summary indexes can readily be presented in verbal text or tables, the display seems wasted if it does not show the individual data points and if the summary does not indicate spread of the data rather than indexes for stability of the mean.

If the visual goal is really stochastic, a separate problem arises from the choice of a standard error. For a stochastic Z (or t) test, or for a confidence interval, the statistical formula uses a standard error calculated from the combined (or “pooled”) variance in the two groups. Thus, for purely stochastic displays, each group should have an identical standard error, derived from the combined variance. If the standard errors are shown individually for each group, they might be used for the type of crude mental evaluations described in Section 13.3.2, but the exact sizes of the standard errors are often difficult to discern from the graphical display.

Probably the main value of showing standard errors is to “market” the results by disguising spread of the data. Because the standard deviation is divided by n, the standard error is always smaller than the SD and makes things look much more compact, tidy, and impressive. The SE becomes progressively smaller with enlarging sample sizes, thus allowing two big groups of data with a dismayingly large amount of overlap to appear neatly separated.

If aimed at the appropriate goal of descriptive communication, the display of individual points will often suffice to show the data, because the corresponding summary values will usually appear in text or tables. An appropriate choice of summaries, however, can be valuable for descriptive (and even stochastic) evaluations of “bulk,” as discussed earlier. The summaries are also necessary for large data sets having too many points to be shown individually. The comparison of two box plots is the best current method of showing these summaries. With suitable artistic ingenuity, the individual points might

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even be shown inside each box, and they could replace (or be shown alongside) the “whiskers” beyond the box.

Some good and bad examples of two-group dimensional displays are shown in the sections that follow. Because some of the displays will be adversely criticized, the sources of publication are not identified, and the names of some of the original variables have been changed.

16.2.2Conventional Displays of Individual Data Points

This subsection contains displays of various conventional arrangements of the points of data for two

if each person were shown with a dot, group means with

lines, and special individuals identified with a label on the graph and an arrow pointing to the dot.

In Figure 16.21, a logarithmic scale is used to cover the wide range of values for data points shown in a single vertical array for three groups, with different symbols. The three groups are so clearly separated that no summary expressions are needed to emphasize the distinctions. In Figure 16.22, data points are also shown for three groups, together with indications of mean and SEM for each group. In the original report, no stochastic calculations were presented for the textual claim that the special amine was “consistently raised in uraemia,” but the separation of the data points indicates that no such calculations were needed. The display of SEM values thus seems redundant, particularly as no claims were made for stochastic distinctions between the “blind-loop” and “control” groups.

Figure 16.23 is remarkable for displaying the summary results with medians and an inner 80-per- centile zone rather than with Gaussian indexes. The authors13 say they wanted to avoid Gaussian assumptions and that “this type of presentation is easily understood by most people.” The research was used, however, to establish a zone of normal values in 704 “healthy asymptomatic aircrewmen.” The 10th and 90th percentile boundary points were chosen “arbitrarily … [as] conservative reference values for the response of healthy men to treadmill exercise” and also “to exclude individuals having subclinical

© 2002 by Chapman & Hall/CRC

© 2002 by Chapman & Hall/CRC

conditions.” Despite the excellent clarity of the visual display, many clinicians will have doubts about establishing a “range of normal” by arbitrarily excluding 20% of apparently healthy persons.

16.2.3Bar Graphs

For large groups, which may have too many individual data points to be easily displayed, the results are often shown with bar graphs. The lengths of the bars show the means, and flanges are added to show dispersion (usually SE). Figure 16.24 gives an example of this tactic for comparing results, in two groups of about 48 animals marked with cross-hatched or black bars, at different times after special administration of insulin. The symbols “N.S.” and “p < 0.05” are added to show the absence or presence of stochastic significance at each time point of comparison. This type of graph offers little that cannot be discerned from the data summaries for each contrast, but a visual display may be valuable for showing the trend of results over time.

A particularly complex bar chart, which probably tries to do too many things, is shown in Figure 16.25. The lengths of the bars indicate the means, but the quartile and median values are appended, together

© 2002 by Chapman & Hall/CRC

© 2002 by Chapman & Hall/CRC

with a special symbol (∆ ) for demarcating the increment in means. This same information could probably have been communicated, with greater artistic ease and visual effectiveness, in two contrasted box plots. Note that the half-H-spreads do not overlap, consistent with a statement in the text that the 73-day difference in medians is “highly significant with non-parametric tests.”

16.2.4Undesirable Procedures

The next set of illustrations shows some particularly unattractive or undesirable procedures. In Figure 16.26, information that could readily have been understood with simple, direct summaries has been displayed in bar graphs. Beyond the basic redundancy, the graphs do not indicate the group sizes or the identity of the top flanges as either SD or SEM. The artist has also “volumized” the bars with an unnecessary third dimension. As a final undesirable touch, the legend erroneously states that the bar graphs are histograms.

Figure 16.27 is another volumized bar graph, which also omits showing group sizes. The unidentified “SED” at the far right of the graph is presumably standard error of the difference, explained in the text only as having been obtained “from the error mean squared of the respective ANOVA (analysis of variance) table.” Aside from the flaws in display, however, the data themselves may not have been appropriately analyzed. According to the text of the report, all the results were obtained in two sets of crossover studies of 6 patients. In one crossover, the patients went from a supine to a tilted position, and in each position they were untreated or received a particular treatment. The research was thus aimed mainly at contrasting the two sets of changes in a single group, but the results are presented as though four groups were being compared.

Figure 16.28 shows two of several similar structures displaying blood pressure and heart rate in response to five different treatments given to 14 patients. The original legend listed only the abbreviations used for the treatments and had no other identifying information. What at first seems to be an incomplete box plot for the blood pressure values could eventually, by careful reading of the text, be discerned as bars having mean systolic pressure on the top of each “box” and diastolic pressure on the bottom. The flanges represent standard deviations. In the lower display for heart rate, the circles are presumably central indexes, whose actual value is effectively obscured by the circles, and whose identity — as means or medians — is not listed in either the original legend or the original text.

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